it is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. the proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.

The defining ideal $I_{\bb X}$ of a set of points $\bb{X}$ in $\bb{P}^n_1 x \ldots x \bb{P}n_k$ is investigated with a special emphasis on the case when $\bb{X}$ is in generic position, that is, $\bb{X}$ has the maximal Hilbert function. When $\bb{X}$ is in generic position, the degrees of the generators of the associated ideal $I_{\bb X}$ are determined. $\nu(I_{\bb X})$ denotes the minimal number of generators of $I_{\bb X}$, and this description of the degrees is used to construct a function $v(s;n_1,\ldots,n_k)$ with the property that $\nu(I_{\bb X})\,{\geq}\, v(s;n_1,\ldots,n_k)$ always holds for $s$ points in generic position in $\bb{P}n_1 x \ldots x \bb{P}^n_k$. When $k\,{=}\,1$, $v(s;n_1)$ equals the expected value for $\nu(I_{\bb X})$ as predicted by the ideal generation conjecture. If $k\,{\geq}\, 2$, it is shown that there are cases with $\nu(I_{\bb X}) > v(s;n_1,\ldots,n_k$). However, computational evidence suggests that in many cases $\nu(I_{\bb X})\,{=}\, v(s;n_1,\ldots,n_k$).

The classical Ramsey theorem states that every graph contains either a large clique or a large independent set. Here similar dichotomic phenomena are investigated in the context of finite metric spaces. Namely, statements are provided of the form ‘every finite metric space contains a large subspace that is nearly equilateral or far from being equilateral’. Two distinct interpretations are considered for being ‘far from equilateral’. Proximity among metric spaces is quantified through the metric distortion $\alpha$. Tight asymptotic answers are provided for these problems. In particular, it is shown that a phase transition occurs at $\alpha=2$.

Our object of study is the natural tower which, for any given map f[ratio ]A→B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E∗, yielding, for example, an analogue of Quillen's plus-construction for E∗. We discuss in detail the case of ordinary homology with coefficients in ℤ/p or ℤ[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f[ratio ]A→pt), which allows us, among other things, to generalize Neisendorfer's observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M(ℤ[1/p], 1).

Given a family of sets/vectors of the same cardinality/dimension you get the shadow by deleting one element/coordinate from a set/vector in all possible ways. You find the family with the smallest shadow by ordering all sets/vectors. The set case was solved by Kruskal (1963), and Katona (1966), and has many applications. We study two orderings which solve the 0, 1 vector case, and give the shadow size.

Various kinds of minimal 3-spheres with constant sectional curvature in the complex projective space $\mathbb{C}^n$ are studied.

Fix an integer $N\ge 2$. For a positive integer $n\in\Bbb N$, let $n=d_{0}(n)+d_{1}(n)N+d_{2}(n)N^{2}+\ldots+d_{\gamma(n)}(n)N^{\gamma(n)}$ where $d_{i}(n)\in\{0,1,2,\ldots,N-1\}$ and $d_{\gamma(n)}(n)\not=0$ denote the $N$-ary expansion of $n$. For a probability vector $\bold p=(p_{0},\ldots,p_{N-1})$ and $r>0$, the $r$ approximative discrete Besicovitch–Eggleston set $B_{r}(\bold p)$ is defined by $$B_{r}(\bold p) = \bigg\{n\in\Bbb N\,{|}\,\bigg\|\,\frac{|\{0\le k\le\gamma(n)\,{|}\,d_{k}(n)=i\}|}{\gamma(n)\,{+}\,1} - p_{i} \bigg| \le r \,\,\text{for all $i$} \bigg\}\,,$$ that is, $B_{r}(\bold p)$ is the set of positive integers $n$ such that the frequency of the digit $i$ in the $N$-ary expansion of $n$ differs from $p_{i}$ by less than $r$ for all $i\in\{0,1,2,\ldots,N-1\}$. Three natural fractional dimensions of subsets $E$ of ${\mathbb N}$ are defined, namely, the lower fractional dimension $\underline\dim(E)$, the upper fractional dimension $\overline\dim(E)$ and the exponent of convergence $\delta(E)$, and the dimensions of various subsets of ${\mathbb N}$ defined in terms of the frequencies of the digits in the $N$-ary expansion of the positive integers are studied. In particular, the dimensions of $B_{r}(\bold p)$ are computed (in the limit as $r\searrow 0$). Let ${\bold p}=(p_{0},\ldots,p_{N-1})$ be a probability vector. Then $$\lim_{r\searrow 0}\,\,\underline{\dim}(B_{r}({\bold p})) = \lim_{r\searrow 0}\,\,\overline{\dim}(B_{r}({\bold p})) = \lim_{r\searrow 0}\,\,\delta(B_{r}({\bold p})) = -\frac{\sum_{i}p_{i}\log p_{i}}{\log N}\,.$$ This result provides a natural discrete analogue of a classical result due to Besicovitch and Eggleston on the Hausdorff dimension of certain sets of non-normal numbers.

Several applications to the theory of normal numbers are given.

In the last decade much research has been done on questions related to circle packings, which are collections of circles in the plane with prescribed patterns of tangencies encoded by simplicial 2-complexes. The interest in circle packings and their connections with analytic functions was initiated by W. Thurston in 1985 when he suggested a method for approximation of conformal mappings using circle packings [20]. This method was shown to work by B. Rodin & D. Sullivan [15] later that year (see also [17]). Since then several results have been published (for example, [3, 4, 5, 10, 11, 14]) which seem to indicate the possibility of creating a discrete analog of the theory of analytic functions via circle packings. However, most of these results deal with univalent circle packings, that is, genuine packings of circles. Influenced by the idea of [3] that circle packings might be used to construct discrete parallels of analytic functions, we started in [7] to investigate non-univalent circle packings, particularly branched packings. We introduced there the notion of discrete Blaschke products and proved a branched version of the finite Riemann mapping theorem (see [20, 15]). We feel though that to have a solid foundation for a circle packing analog of the theory of analytic functions one needs something more. Since complex polynomials form a fundamental and very important class of analytic functions, it is natural to try to establish circle packing counterparts of these mappings. This is exactly the main theme of our paper.

Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be $\ell$-embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups.

THEOREM. Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be$\ell$-embedded in a finitely presented lattice-ordered group.

If $\xi$ is a real number, let $D(\xi)$ be the Abelian rank 2 group $\Z^2$ with order $(m,n)>0$ if and only if $m+n\xi>0$.

COROLLARY. $D(\xi)$can be$\ell$-embedded in a finitely presented lattice-ordered group if and only if$\xi$is a recursive real number.

Thus an algebraic characterisation of recursive real numbers is obtained. In particular, $\pi$ is ‘$\ell$-algebraic’ in that it can be captured by finitely many relations in this language.

The heat content asymptotics on a Riemannian manifold with smoooth boundary defined by Dirichlet, Neumann, transmittal and transmission boundary conditions are studied.

Let P be a polyhedron in H3 of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of Isom H3. Let Γ+(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ+(P) is a Kleinian group of finite covolume. Γ+(P) is called a polyhedral group.

As discussed in [12] and [13] for example (see §2 below), associated to a Kleinian group Γ of finite covolume is a pair (AΓ, kΓ) which is an invariant of the commensurability class of Γ; kΓ is a number field called the invariant trace-field, and AΓ is a quaternion algebra over kΓ. It has been of some interest recently (cf. [13, 16]) to identify the invariant trace-field and quaternion algebra associated to a Kleinian group Γ of finite covolume since these are closely related to the geometry and topology of H3/Γ.

In this paper we give a method for identifying these in the case of polyhedral groups avoiding trace calculations. This extends the work in [15] and [11] on arithmetic polyhedral groups. In §6 we compute the invariant trace-field and quaternion algebra of a family of polyhedral groups arising from certain triangular prisms, and in §7 we give an application of this calculation to construct closed hyperbolic 3-manifolds with ‘non-integral trace’.

A coherent system ([Escr ], V) consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Motivated by the usual notion in geometric invariant theory, a notion of slope stability can be defined for such objects. In the paper it is shown that stability in this sense is equivalent to the existence of solutions to a certain set of gauge theoretic equations. One of the equations is essentially the vortex equation (that is, the Hermitian–Einstein equation with an additional zeroth order term), and the other is an orthonormality condition on a frame for the subspace V⊂H0([Escr ]).

The main result ensures that the scalar problem $x''\,{=}\,f(x)$, $x(0)\,{=}\,x_0$, $x'(0)\,{=}\,x_1,$ has a nonconstant locally $W^{2,1}$ solution if and only if there exists a nontrivial interval $J$ such that $x_0 \in J$, $f \in L^1_{\rm loc}(J)$, $x_1^2+2\int_{x_0}^{y}{f(s)\,ds} > 0$ for almost all $y \in J$ and \[\frac{\max\{1,|f|\}}{ \sqrt{x_1^2+2\int_{x_0}^{\cdot}{f(s)\,ds}}} \in L^1_{\rm loc}(J).\] Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.

We consider the Sturm–Liouville equation \renewcommand{\theequation}{1.\arabic{equation}} \begin{equation} y^{\prime\prime}(x)+\{\lambda - q(x)\}y(x) = 0\quad (0 \le x < \infty) \end{equation} with a boundary condition at $x = 0$ which can be either the Dirichlet condition \begin{equation} y(0) = 0 \end{equation} or the Neumann condition \begin{equation} y^\prime(0) = 0. \end{equation} As usual, $\lambda$ is the complex spectral parameter with $0 \le \arg \lambda < 2\pi$ , and the potential $q$ is real-valued and locally integrable in $[0, \infty)$ .

Let k be an algebraically closed field of characteristic p > 0, and let G be a connected, reductive algebraic group over k. In [8] and [11], conditions on the dimension of rational G modules were seen to imply semisimplicity of these modules. In [8], certain of these conditions were extended to cover the finite groups of Lie type. In this paper, we extend some of the results of [11] to cover these finite Lie type groups. The main such extension is the following result.

Let $p$ be a prime and let $q = p^a$, where $a$ is a positive integer. Let $G = G({\mathbb{F}}_q)$ be a Chevalley group over $\mathbb{ F}_q$, with associated system of roots $\Phi$ and Weyl group $W$. Steinberg showed in 1957 that $G$ has an irreducible complex representation whose degree equals the $p$-part of $|G|$ [11]. This representation, now known as the Steinberg representation, has remarkable properties, which reflect the structure of $G$, and there have been many research papers devoted to its study. The module constructed in [11] is in fact a right ideal in the integral group ring $\mathbb{ Z}G$ of $G$, and is thus a $\mathbb{ Z}G$-lattice, which we propose to call the Steinberg lattice of $G$. It should be noted that lattices not integrally isomorphic to the Steinberg lattice may also afford the Steinberg representation, and such lattices may differ considerably in their properties compared with the Steinberg lattice.

In an attempt to prove all the identities given in Ramanujan's first two letters to G. H. Hardy, G. N. Watson devoted a paper to the evaluation of G1353. At the end of his paper, Watson remarked that his proof was not rigorous as he had assumed certain identities (see (1.1) and (1.2)) which he found empirically. In this paper, we use class field theory, Galois theory and Kronecker's limit formula to justify Watson's assumptions. We shall then use our results to compute some new values of Gn.

The minimum number of critical points of a small codimension smooth map between two manifolds is computed. Some partial results for the case of higher codimension when the manifolds are spheres are also given.

An explicit geometrical study of the curves

[formula here]

is presented. These are non-singular curves of genus 3, defined over ℚ(a). By exploiting their symmetries, it is possible to determine most of their geometric invariants, such as their bitangent lines and their period lattice. An explicit description is given of the bijection induced by the Abel–Jacobi map between their bitangent lines and odd 2-torsion points on their jacobian. Finally, three elliptic quotients of these curves are constructed that provide a splitting of their jacobians. In the case of the curve [Cscr ]1±√2, which is isomorphic to the Fermat curve of degree 4, the computations yield a finer splitting of its jacobian than the classical one.

The commonly known properties of continuous measures such as the symmetry of the A∞ condition, its equivalence to the reverse Hölder inequality, the left-openness of the Ap condition, etc., are no longer necessarily true when the underlying measure is allowed to have atoms. The measures that preserve these properties are called good measures. The class of good measures is investigated and various criteria for a measure to belong to this class are presented.